Monday, June 1, 2015

MATHS STRAIGHT IN archive from 1st June 2015


Ongoing re-writes, updates and additional material are noted on the LATEST UPDATES page.



NAE  - National Average Earnings / Incomes
AEG% p.a. - Average (National) Earnings Growth% p.a.
TRUE INTEREST -  the marginal rate of interest above AEG%.
WEALTH PRESERVATION - keeping pace with AEG.
WEALTH TRANSFER RATE - the true rate of interest applied to the sum involved.

NOTE: IF A  SAVINGS ACCOUNT ONLY KEEPS PACE WITH PRICES AND IF PRICES RISE LESS QUICKLY THAN NAE, then for example, 20 NAE saved (say half a life-time's income), and invested to keep pace with prices inflation may lose 95% of its value. It could end up as just 1 NAE after a century. This assumes that prices fall behing rising incomes at a rate of 3% p.a.

To preserve the NAE (the half lifetime's income), the 20 NAE, the fund has to grow as fast as NAE grows. That is, at a rate of AEG% p.a.

A zero true rate of interest is a nominal interest rate of AEG%.  A positive true rate of interest of 1% = AEG% + 1%.

That positive true rate of 1% is the rate at which more NAE are added to the fund. If the fund is a loan, that 1% true interest rate is the rate at which NAE are transferred from the borrower to the lender.

How AEG is to be measured.
AEG% p.a. is the rate of growth of NAE. The real question is how NAE is measured. There are people who seem to be getting a larger share of the National Income every year, and those probably do not need to buy a house using borrowed money.

For now, the term NAE will be used as if it represents the average borrowers' rate of earnings increase. If lenders prefer another definition, such as the median level of Earnings, that is open to them to do so.

At the same time, the rate of interest which market forces demand may be more influenced by NAE. Whatever the outcome the mathematics allows for such differences in opinion or differences in the make-up of the index.

How the true rate of interest should behave in theory and in practice has been discussed on various pages of the Main Website.
 This is one such page. Also read the page before it and the page after it. In an undistorted economy the true rate would probably be quite steady. As things are, it is not. How should we cope with that?

The quest to assist all costs, prices, and values, in the economy to respond one-on-one to changes in the level of aggregate demand, even if that is not done with absolute precision, is greatly assisted by this research.

One of the first laws of economics taught is that prices should be able to respond to changes in the level of demand, like rentals and most prices do. Doing this keeps the balance in an economy. 

Unfortunately some of the current economic framework, like housing finance, fixed interest debt, and even interest rates (the price of money), does not comply with this basic law. These costs and prices are forced to move all over the lace. That is because the contracts and ways that we are doing things gets in the way and spoils everything for everyone. Thus most of our costs, prices, and values are constantly out of balance and unpredictable. It makes the economy unpredictable.

In order to maintain a balanced economy nominal interest rates need to respond one-on-one to changes in the level of aggregate demand. Taking the growth rate of National Average Earnings, (NAE) and National Average Earnings Growth, (AEG% p.a.), as a proxy for the rate of growth of aggregate demand per person enables us to research the relevance of this measure to the setting of interest rates (a price).

And it enables us to derive a new model for lending and savings in which the cost of repayments, the rate of interest, and the amount which can be lent safely, all begin to look consistent with the quest to allow economies to self-adjust through compliant cost, price, and value, adjustments. This page sets out the mathematics which enables this to become practical in the area of lending and savings.

Whereas mortgage repayment costs are currently over-sensitive to interest rate changes, (they do not respond one-on-one to changes in AEG% p.a. but nearer to 10:1), this places the entire dynamics of economies in disarray.

The various linked websites deal with this in considerable and practical detail. 

Notable links are:

Following on from that is the page on QE AS IT SHOULD BE, which has since been adopted by a group of prominent economists in a letter to the Financial Times. But the IngramSure research group has pointed out some shortcomings of that suggestion as made by those economists. 

  • The new savings and debt models invented by the IngramSure research group are needed to safeguard against too much money being printed,

  • The stimulus should be equally to all spending levels throughout the entire economy, as far as practical. 

  • To be a stimulus it should be for a short and pre-defined (announced) duration. This we are all agreed upon.

  • New money created should not be for government use - a moral hazard.

WEALTH BONDS describes how the mathematics on this page can be used to create a good and compliant (with first principles of macro-economic design) savings and debt structure for government borrowing. It is similar in merit to Robert Shiller's Trills (GDP-linked Bonds), but wealth bonds eliminate and overcome the non-compliant disadvantages of trills:

GDP-linked Bonds for government use only will not provide a level playing field for borrowing which includes the private sector.

GDP-linked Bonds are not compliant in responding to AEG% p.a. which private sector borrowing needs to comply with in order to raise funds cheaply - the demand is there.

The outcome:
  • a limited market, 
  • a distorted investment, 
  • an unbalanced borrowing framework for the nation.

The general principles of good macro-economic design in the lending and savings sector:
  1. All costs, prices, and values should be able to adjust to changes in NAE / AEG% p.a. as the nearest convenient proxy for the changing level of aggregate demand per person.
  2. There should be a level playing field between public and private sector finances.
  3. There should be a level playing field in respect of the taxation of all investments, giving tax exemption on gains up to the level of AEG% p.a.
  4. No tax relief on interest up to the level of AEG% p.a.
The reason for these tax rules (items 3 and 4 as listed) is that any gains up to the level of AEG% p.a. would be a tax on wealth.

Example 1: a fund of 20 NAE which keeps pace with prices might lag behind incomes growth (AEG% p.a.) by 3% p.a. over a century. At the end of that time the fund would have only 1 NAE remaining. A 95% loss of wealth compared to the original expectations.

Example 2. A government of a nation of 10 working people, could take a percentage of income from those 10 people and pass it to the 4 people who had retired as their pensions. Those pensions would keep pace with NAE.


  • A fund that is linked to the growth rate AEG% p.a. should not be taxed on capital gains or interest added to capital up to the level of AEG% p.a.  

  • Tax relief should not be needed or given.

  • Preserving wealth means keeping pace with AEG% p.a.



The main equation is extremely simple:

P% = C% + D% + I%       -   (i)

and its derivation, based on a Risk Management Chart, is unbelievably simple. Let's start the derivation right now. 

Please don't worry about the exact definition of I% or AEG upon which it is based. We will come to that later and I will explain why I% is absolutely important as soon as it seems right to do so.



DEFINITIONS  - refer back to these when necessary.
P% p.a. = P p.a. / L x 100% 

I often drop the 'p.a.' part as this is understood.

P% = P / L x 100% 

is easier to read.

P  = the current level of annual payments p.a.

L = the debt / loan / mortgage value
So 'P%' is the percentage of the debt 'L' that is being paid by money payments 'P' p.a. 

'P' is not all capital repayment, some of it is interest.

r% = Nominal Interest.

If P% = r% then this is an interest only payment. No capital is repaid. 

For example,
 if P = 7 p.a. and L = 100, then 
P% = 7% p.a. 

If nominal interest rate r% = 7% and P% = 7% then only the interest is being paid.

e% p.a. = the percentage rate at which debt servicing payments 'P' p.a. increase every year.
  P = the annual amount of MONEY paid. Remember 'P' is NOT 'P%'.

 if P = 100 and e% = 3% p.a.  then in the following year the value of P = 103. Any currency.

AEG% p.a. is an index of Average Earnings Growth - something that needs careful thought as discussed on the page entitled MATHS 3 - defining AEG/GDP

Now we can draw this chart below whereby the position of the 'X' can be placed anywhere so as to describe the current state of any repayments schedule for any debt - mortgage, bond, overdraft, or otherwise. 

regular payment debts have a P% and an e% at all times. So every kind of  such debt repayments schedule has a place on this chart. See FIG 1:

FIG 1  Ingram's Risk Management Chart
The chart has two Axes, two straight dashed lines, one at 45 degrees slope representing a standing loan line and one vertical line representing the current rate of AEG% p.a.,  and an 'X' whose position on the chart describes the current state of any regular repayments loan. 

The position of the 'X' describes the current value of P% and of e%. And the position of the 'X' has a visible horizontal distance from the AEG% p.a. line and a visible vertical distance from the standing loan line - two safety margins to be aware of.

 'X' has a value of e% and a value of P% - as any debt repayments schedule has. Any place on the chart has a value for P% and a value for e%, both being p.a.

If placed at the origin, the position of the 'X'  would describe an overdraft: no payments, and no escalation at e% p.a.

In this example the position of the 'X', being to the left of the AEG% p.a. vertical line, describes payments that are rising more slowly than AEG% p.a. (incomes) making it easy on borrowers, and keeping arrears rates low.

Because the 'X' lies to the left of the AEG% p.a. line and because the 'X' is above what I will explain later is a standing loan line, the debt is being repaid. This is the safe area of the chart. 

If the 'X' is placed anywhere on the Y-Axis above the standing loan line, and if the 'X' is pinned permanently to the Y-axis, it is a 'Level Payments' Loan: 
 e% = 0%.  I.e. no e% p.a. escalation of the payments is scheduled.

With a Level Payments Mortgage / Loan when the interest rate rises then the payments jump to a new level. The change is not scheduled and it may create arrears problems.

In this Level Payments (LP) mortgage model, the distance of the 'X' from the AEG% p.a. line depends upon where the AEG% line happens to be at any one time. 

When the 'X' is to the left of the AEG line the payments are getting less as a proportion of the borrowers' incomes - at least on average and assuming that the AEG line represents the average borrower. We can discuss the case when The average borrower is not fully represented by the AEG line later.

If average incomes are not rising much, then the stress on borrowers can be considerable. The LP Mortgage Model relies upon incomes to be rising at a reasonable rate to ease the cost burden on borrowers, especially in the early years. If incomes are falling, (if the AEG% p.a. line moves to the left of the Y-axis), the stress becomes unmanageable. You get very high levels of non-performing loans, just like in Europe now.

Instead of payments falling as a % of an average annual income' like this sketch:

NOTE: The 21% of income line represents a possible rental which may be rising at the same pace as average incomes or AEG% p.a.

or like one of these downwards sloping lines:

Instead, the cost might be rising as a '% of average income' more like this:

 If the 'X' is attached to the AEG% p.a. line (which moves) it is a wages / incomes-index-linked loan such as is currently in use in Turkey. In Turkey, both the interest rate and the annual payments increases, are equal to the selected AEG% p.a. index, which in Turkey is the wages index of Civil Servants. So e% = AEG% p.a. at all times. And the interest rate (or indexation of capital) is at the same rate plus any additional interest added on top.

This is not a very comfortable arrangement for borrowers. It is like paying a fixed interest mortgage when average incomes are not rising or falling. Or it could be like paying a high level of rental. Rentals usually cost less than first year mortgage costs. And rentals may rise at a rate similar to average incomes as people compete to rent a property.

The 'X' needs to be to the left of the AEG% p.a. line, not attached to it so as to give relief and to allow for people whose incomes are not rising at the average rate.

Wherever the 'X' is placed, the position of the 'X' describes the scheduled current condition of a particular debt repayment model.

The standing loan line is a line which, if the 'X' is placed anywhere on it, the loan will never be repaid. If the 'X' happens to be on the standing loan line then the value of the debt 'L' will be rising at the exact same e% p.a. pace as the payments 'P' are  increasing. 

The ratio P/L and the value of P% p.a. never changes. Normally P% p.a. rises every year as the loan gets repaid.

It is like a race in which the rate of payments increase 'e% p.a.' is chasing the loan size 'L' increase, which is also rising at 'e% p.a.'

Neither side wins. It is a dead heat. 

The same thing can happen when incomes and payments are both falling. In this case we have to extend the standing loan line to the left of the Y-Axis:

In the case where the 'X' is above the standing loan line the mortgage is being repaid even though the payments are falling every year. In this example, the level of payments 'P' is about normal because of the very low nominal rate of interest of 3%.

This spreadsheet (at right) applies if we move the AEG% p.a. line onto the Y-axis.  That would mean that average incomes are not rising or falling, but the payments are falling and the debt is falling even faster as shown in the table. 
The interest rate is as shown in the risk management chart above at 3%. To simplify the table, all transactions occur at year's end.

On this schedule the payments are falling at 4% p.a.


To find out where that standing loan line may be we have to do some thinking.

What we just said in mathematical terms is this:

For a standing loan to exist, P / L = Constant. 

The ratio never alters. Both variables 'P' and 'L' grow or fall at the same percentage pace, 'e% p.a.'. 

For example, if P% = r% and e% = 0% it is a traditional interest-only 'standing loan'
 in which case both 'P' = constant, and 'L' = constant. No capital is repaid. Only interest is paid. P/L = Constant.

The 'X' is then on the Y-axis at P% = r%.

If P% is 1% less than the interest rate r% then what?

Then the loan size 'L' will be higher by 1% at the end of the year. 

So the payment 'P' must increase by 1% to draw level so as to keep that ratio of 'P / L' unchanged. At year end, after 'L' has increased by 1%  'P' has to catch up with 'L'. So e% = 1% p.a.
 See ‘X1’ in Fig 2

FIG 2 - Standing Loan when e% = 0% and e% = 1%

You can do the same for P% = 2% less than r%, so that 'L'
finishes the year 2% higher and then 'P' has to rise by 2% to keep pace... FIG 3:

FIG 3 Standing Loan at e% = 0%, 1%, and 2% p.a.

...and so on for all values of P% and e%.

Thus the standing loan line falls 1% for every 1% it moves to the right. 

Every time P% drops by 1%, e% p.a. has to rise by 1%.

Hence the Standing Loan Line forms an equilateral right angled equal sided triangle with the two major axes, and with the standing loan line as the hypotenuse. The standing loan line crosses both the Y-axis and the X-axis at the same value,  r%.

The full algebraic derivation of this line can be found in the Appendix for Standing Loan Line.

To make a loan affordable and safe the 'X' has to be sufficiently above the standing loan line so that it gets repaid and it has to be sufficiently to the left of the AEG% p.a. line so that the payments increase more slowly than incomes do.

This way it gets easier for most people to repay every year and the arrears rate is low. This is a key feature which Turkey has overlooked.

Since no individual's income rises at the exact AEG% p.a. rate, the 'X'must be placed some distance to the left of the AEG% p.a. line if stress / payments fatigue is to be avoided by the majority (say 95% or more) of borrowers.

To estimate how far to the left the 'X' should be, I noticed that in the UK when average incomes were growing at around 4% p.a. there did not seem to be a problem with arrears rates. So I have adopted that 4% as a standard in my illustrations. Whether that works for all nations and whether it is the ideal rate is not something to judge until it has been put to the test in practice. But it looks like somewhere to start.

If the payments do not get easier every year, that is if the 'X' is on the AEG% p.a. line, you have the Turkish Wages-Linked Mortgage Model where the payments increase at the same rate of increase as the wages index that they use. This creates 'Payments Fatigue' and limits the amount that can be lent safely. Because not all incomes rise as fast as the index this is likely to increase the level of arrears and defaults which otherwise may be close to zero.

In PART 2 below, will derive an equation that calculates the value of P%. This is the key equation that we need.

So far we have established the framework of the Risk Management Chart, the fact that we can draw a standing loan line which slopes downwards at 45 degrees, and crosses both Axes at r%, the level of the nominal rate of interest.

This means that the vertical and horizontal sides of the triangle formed with the two major axes, have equal sides, r%.

We have also placed an AEG% (Average Earnings / Incomes Growth) line on the chart. We know that this line moves to the left and to the right as the value of AEG% p.a. changes.

We know that we do not want the AEG% p.a. line to get too close to the 'X', especially in the early years. And we want the 'X' to be far enough away from the AEG% p.a. line to be of use to borrowers whose incomes are not rising as fast as average.

If for some reason AEG% p.a. does not represent the average borrower but is an index that rises faster or slower than it should do in the circumstances then a line to the left or to the right of the AEG% p.a. line can be used in its place. The maths of risk of arrears and default remains the same.

Derivation of the Main Equation

All Ingram Savings and Lending (ILS) Mortgage Models insert a figure, 'D%' into what I call my safe entry cost equation:

P% = C% + D% + I% 

P% is the entry cost - the first year repayments level. But it can also be the equation for any year. The point here is that the first year is when the loan: income ratios are highest and the risk of a problem going forward is the greatest. So we need to have an adequate set of margins of safety at this point. Those margins are 'D%' and 'C%'.

They form the buffer if 'I%' rises.

'D%' is the distance of the 'X' from the AEG% p.a. line. The larger the value of 'D%' (the greater the distance the 'X' is to the left of the AEG% line), and the faster the payments get easier as a prortion of the 'average' borrowers' incomes. 

'D%' is called the rate of 'Payments Depreciation'.

C% = the distance of the 'X' above the standing loan line. It is a form of capital repayment.

I% = r% - AEG%  - it is the marginal rate of interest above AEG%. r% is the nominal rate of interest.

r% = AEG% + I% might explain that better. It is the same equation turned around. This equation says that the nominal rate of interest can be split into two parts - the rate of AEG% p.a. and the additional, or marginal rate above AEG% p.a.

I gave I% a name because it is a key element of the main equation. I called it the true rate of interest. The following maths will show why that is so.

All are p.a.

FIG 4 - All Three Variables Displayed

Here I have added those three new letters to the chart.

NOTE the precise definition of D%: 
D% = AEG% - e%     - (ii)   - all are p.a.

Or if you prefer this can be re-arranged:

e% = AEG% - D%

Meaning that the annual rate of increase in the payments is D% less than AEG% p.a.

I tell people that the nominal rate of interest has two components for these purposes:

The AEG% p.a. component of r% preserves the value of a debt (or of savings) as a multiple of National Average Earnings (like average incomes). If NAE increases at g% p.a. then National Average EArnings Growth (AEG% p.a.) is g%. Substitute any number for the letter 'g'. It makes sense.

If the rate of interest r% being added is the same as the rate at which average incomes / NAE is growing, then the loan / income ratio is not changing. Incomes and loans are all rising (on average) at the same pace.

I got to use NAE as a result of not finding an index of average incomes when these studies forst began. There was an index of Average Earnings, so I used that to test the theory. Test results were very good.

The true interest rate ‘I%’ is the marginal rate above AEG% p.a.

As already stated above, by this definition:

r% = AEG% + I%
or we can write:

I% = r% - AEG% ............(iii)

It is the same for governments - the 'debt / GDP' ratio is a similar concept, with GDP being the national income. It is preferred that this ratio falls rather than rises over time if the intention is to repay the government's debt, although for a limited time it may be considered alright to just keep that ratio constant or increasing.

In the case of a government debt that limited time of keeping the ration constant or fairly stable may not be so limited - it can be a very long time or even for ever as long as the government is not borrowing too much and preventing others from getting their share of the funds available for borrowing.

NOTE: I will return to this topic in another paper that I am writing on money supply and the management of money supply. 


If r% = AEG% p.a., 

then if no payments 'P' are made, the value of the loan 'L' would increase at the same pace as Average Earnings / Incomes (AEG% p.a.).

But if r% = AEG% + I% then the debt would rise 'I% p.a'. faster than average earnings / incomes. That is, 'I%' faster than AEG% p.a. That would be like a government rolling up interest that was greater than the rate of growth of GDP.

So we have three variables to consider: 'C%', 'D%', and 'I%', as seen in FIG 4 above and again in FIG 5 below. 

In FIG 5 we link all three variables together with equilateral triangles, like this:

FIG 5 - Derivation of the General Equation for P%
Note that I have constructed two small equal-sided right angle triangles: 

One with two equal sides of length I%,
One with two equal sides of length D%.

Hence by adding up the vertical distances we can see that:

P% = C% + D% + I% - Ingram’s Safe Entry Cost Equation

Although this equation can be used to estimate a safe entry cost by ensuring that the variables 'C%' and 'D%' are large enough at outset, this equation has a general use in analysing the component parts that make up the value of ‘P%’ for ANY repayments schedule for ANY debt.

This means that ALL debt repayment schedules have these three parts, whether intentionally or not. They all have a value of 'I%' and of 'D%' and of 'C%' at all times. The only difference is that there are different ways to allocate the various values to the three variables.

There is no debt repayments schedule which has no value of 'e%' or no value of 'P%'. Since all debt repayment schedules have a value 'P%' they also have a value for 'C%', 'D%', and 'I%'.

If all three are unstable or if 'C%' is unstable then 'P%' will be unstable. That spells trouble. If 'D% p.a.' is allowed to go negative and stay negative for very long then there will be a lot of arrears.



Given that ‘D%’ needs to be positive, we can decide which debt repayments models are safe and which are not. Those that have no control over ‘D%’ are not safe - at least not for people. (A government may be able to cope with a zero 'D%' because it will not get any payments fatigue). A business may have rising income and rising turnover and profits somehow linked to AEG. In that case there will be no Payments Fatigue. But business loans are usually paid fast so 'C%' is large and that gives plenty of scope for rescheduling if needed by reducing 'C%'.

For example, a Level Payments (LP) Schedule in which e% = 0% until the interest rate changes (if it does), mans that:

D% = AEG%.

Remember, 'D%' is the distance of the 'X' (on the Y-Axis) from the AEG% p.a.' line.

D% = AEG% - e% ..........(ii) above.
where e% = 0%

And as in Europe recently, AEG% can be negative which for a fixed interest or a positive interest rate can be a disaster.

In developing nations that use the Level Payments Model (mostly they do), AEG% can be highly positive. The nominal rate of interest 'r%' is also high. So 'P%' is very high.

P% = C% + D% + I%

P% is high because 'C%' is large.

This means that not much can be lent. This is why I do not like the Level Payments (LP) Model for any kind of economy. At high AEG it is a failure and at low AEG it is dangerous.

When AEG% is negative and r% remains positive, as in Europe recently, we get this:

r% = AEG% + I% 

so if AEG% = -5% say, and r% = 3% say we get:

3% = -5% + I%

So I% = 8%

As can be seen, if AEG% is negative, this raises the value of I% significantly. And THAT is the rate of transfer of spendable income from borrowers to lenders. It is high because average incomes are falling.

The average value of 'I%' for the UK mortgages from 1970 to 2002, averaged 3% p.a. See the Graphs FIGs D and E above, or for the discussion, on this link.

8% p.a. true interest is more than the long term rate of return on USA and UK Equities which have averaged 7% p.a. real return and so maybe only 4% true return (above AEG% p.a.). And remember, investments in property and equities carries a lot of volatility and risk. See this link.

At low interest rates like this, how large would the mortgages be?

In practice, lenders calculate P% using just r% without any regard to either D% or I%. 

As far as they are concerned:
P% = C% + r% - all of the nominal interest plus some capital.

So the smaller r% becomes the more that they say they can lend. What they say they can lend is determined by the value of r%, the repayment term, and the borrower's allowable, or net spendable, income.

Based on their equation, they are saying that all of the nominal interest 'r%' transfers wealth from the borrower to the lender, so all of r% must be paid.

So when 'r%' is very low far too much can be lent, leaving no space for 'D%' and 'I%', neither of which are in the lender’s control. Both are unstable so some allowance for that instability needs to be made. Not doing that is highly dangerous. 

This, and the unemployment to which such oversights contribute when the LP Model feels the strains or crashes, explains why there is a very high level of non-performing loans in Europe.

Check out this table which is  based on what happened to the USA before and after 2007:

If lenders looked at the value of 'P%' and how much space that gave for a decent value of 'D%' 'I%' and 'C%' and if they took note of the mean long term value of 'I%' they would rarely, if ever, use that LP equation for calculating the safe level of repayments or the safe amount of wealth that can be lent.

For example, before the Fed raised interest rates and tried to increase them by 4.5% the value of 'P%' would have been set at 6.1% p.a. for a 25 year mortgage. For a 30 year mortgage 'P%' was 5.4% p.a. and for an interest only mortgage P% = 3.5% p.a.

They were offering interest-only mortgages in 2007.

For a more typical 7% interest mortgage over 25 years, P% = 8.6% p.a and 'C%' = 1.56%

For 30 years P% = 8.1% p.a. and 'C%' = 1.06% and at AEG% = 4% p.a. D% = 4% making I% = 3% p.a. This gives the following options for D% and I%:

P% = 1.06% + 4% + 3% = 8.1% rounded.

Because they us this formula for calculating the amount that they can lend, this leads us all into what I call the Low Inflation Trap, explained on my Blog and again, on South Africa's premier Financial Website, Click here.

If 'I%' rises by 1% then 'D%' would need to fall by 1% to keep 'C%' and 'P%' unchanged. So it is necessary to have a high enough 'P%' at the start, say around 8.5% p.a. so that 'D%' can be positive and there is some space in case 'I%' rises and stays risen.

We will see later that temporary spikes in the true rate 'I%' can be accommodated by the ILS Model, and we will see that the medium term value of 'I%' cannot be much below 2% or above 4% in an efficient economy with efficient lenders and low risk of lending.

The fact that the Fed kept interest rates much lower was unsustainable, and the fact that it is still doing that is also unsustainable. 

In 2007 the rate of inflation was rising so strongly that a 8% value for r% was needed to contain it, but that 8% rate (a rise of 4.5% in the nominal rate) crashed the economy.

The same can happen again as interest rates rise to contain the coming inflation.

That is - UNLESS policy makers DO SOMETHING about their mortgage and government and business debt models.

Something to make them less sensitive to nominal rates of interest whilst still being responsive to true rates of interest.

The AEG% part of the interest can also be replaced with indexation. Adding AEG% interest to a debt is the same as index-linking the capital to AEG% p.a. Both methods increase the value of the debt by AEG% p.a. unless some capital is repaid.

In other words AEG% interest increases the debt at the same pace as average incomes are rising.  If a year's income has been lent then many years later if no payments have been made, a years' income is still owed but it is more in money terms because both debt and incomes will be bigger (or smaller).

For now we are using average incomes, not the incomes of individuals.

If I% = 0% what it means is that all the debts are rising as fast as average incomes, AEG% p.a. It means that average incomes are rising as fast as the debt. This means that the amount of 'average income' that has been borrowed is the same as the amount of 'average income' that gets repaid. I will give a few examples of this and related issues shortly to make it clear.

You can say the same thing about national / government debt. Replace AEG% with the rate of growth of Nominal GDP, (NGDP% p.a.) assuming that tax revenues will rise as fast as NGDP as an approximation. Dropping the 'N' as understood, for GDP, Then:

If 1 GDP is borrowed and if net tax revenues are rising as fast as GDP is rising, then after a year the amount of the debt is still going to be 1 GDP. If this happens then we could say that the Government's marginal (true in this sense) interest rate, I% = 0%. There is no marginal, or additional interest being paid. The nominal interest added is (about) the same as the rate of growth of net revenue income and is the same as the rate of growth of GDP.

We have also seen how to derive the main risk management equations and we have seen that every debt repayment model has the same three elements which together make up the value of P%.

We have seen that the LP (or annuity / Level Payments) mortgage model does not attempt to make the repayments safe and affordable by ensuring that there are appropriate values of C% and D% at all times.

We have seen that the result is that too much can be lent to be safe when nominal interest rates are low, which is usually when inflation rates are low. Low rates create property price bubbles and crashes when interest rates rise. This tendency is much greater than what happens at high inflation and interest rates. That is because the greater the interest rate is initially, the more of 'P%' is made up of interest payments and so a small change in the interest rate has a small effect on 'P%'.

For example:

For example, if interest rates were zero and the mortgage was repayable over an infinite period, (interest only) or almost the same thing, repaying over 100 years, then in theory an unlimited sum could be borrowed. THEN, a 1% rise in the interest rate would almost infinitely increase the cost of the repayments.

The table below shows the sensitivity of an LP 25 year mortgage to a 2% rise in the interest rate. The X-Axis varies the starting level of the interest rate.

When I first did the mathematics I was not really aware of I%. Later, I found that putting I% into the equation simplified everything. At the time, I did not even have the above risk management charts to look at. How I managed to do the maths and get that equation is a mystery to me. In fact, at the time I kept on forgetting how I did it! Now, with the aid of the chart, I can show anyone very quickly how this all works, as shown above.

I was really excited to find that using 'I%' produced such a simple equation. Clearly this was a significant item. There had to be something important about it. If I had not wanted to find a way to make payments get easier every year by linking the repayments to average incomes and at a slower pace, then I would not have found this key ingredient, 'I%'.

I am going to STOP writing this mathematics at this point - to rest.

Later I will add and edit the FIG numbers to the above charts and FIGS.

There is a whole lot more to come including back-testing the ILS Model on past data and doing some standard engineering tests on the responsiveness of the model to extreme conditions.

What follows below is old text which I may edit and retain or not.

More about True Rates of Interest

To see the role that is played by 'I%' let's look at a spreadsheet example. Here I% = 0%:
The loan starts at 100 000 in any currency, adds 5% interest, subtracts a payment and ends with a balance at the end of the year. This balance is taken down to the start of year 2 and the process is repeated.

To make sense of this, the interest is added at year end and that is when the payment is made by which time incomes have risen by AEG% = 5% in this case. As a percentage of that new income the cost of the payment is shown in the final column.

By adding up those final column percentages we find out the cost-to-income of repaying the loan, which comes to one year's income as shown in the bottom line.

We can either add up the money repaid, which is 110,163 as shown and is 10.2% more than was borrowed, or we can add up the '% of income' column which calculates how much income is needed to make the payments.  This shows that 1 year's income was borrowed and 1 year's income was needed to repay that one year's income. Zero cost to income - at least in theory assuming that the income rises at AEG% p.a. all the time.

Because I am using averages here the cost to an individual is not the same as the cost to a borrower whose income rises at the average rate (AEG% p.a.). It is better to say that if 1 NAE (one National Average Earnings / Income) has been borrowed, then it costs the borrower 1 NAE to repay the debt, including the interest of AEG% p.a.

We can regard NAE as units of wealth which the lender has lent and which the lender wants to get back with interest.

In a number of essays that I have written on the website, I explain why there is a link between AEG% p.a. and interest rates. Here is a link to one such essay. I explain why all investments tend to rise at a base rate of AEG% p.a. So any lender wishing to raise funds in competition with those other investments will have to pay an interest rate which has a relationship with AEG% p.a.

Going back to the table above, if you add up the money, it looks expensive: 10.2% more money than was borrowed. But the borrower has not paid any more NAE than was borrowed and the lender has not gained any NAE.

NOTE: The payment is made after the interest is added, both being at year's end by which time the income has risen by AEG%. Hence the fourth year's income is shown.

The net cost-to-income (NAE) is 0% as shown in EXAMPLE 1 above.

EXAMPLE 2 - If I% is positive at 3% we get this after one year:
The amount of income added to the debt in the first year is I% x L, or 3% times three years'  income = 9% of a year's income. The debt is repaid at the end of the first year in this example.

If we say that the loan was 3 NAE then 9% of NAE has been added by the 3% true rate of  interest.

Now we see that maybe it is of use to count cost in terms of income, (or NAE), rather than in terms of money. Food for thought. Something to explore. Let's do that. Let's think about this:

If we look at this from the viewpoint of the lender or the investor in the debt, (think of savers as investors using deposits or think of investors in bonds that are used to provide mortgages), we see from the above example where I% was 3%, that I% is the rate of increase in the debt in units of average income, NAE, or wealth (income) that was lent. 

If r% = AEG% p.a. then the lent income, say 1 NAE, returns as a year's current, spendable income that has been preserved by the addition of interest at the rate at which AVERAGE incomes have been growing.

I will explain that more as we get to more illustrations because usually people do not think of wealth in this way. This is where people get a bit dizzy sometimes. But stay calm, relax, and watch what else the figures will tell us as we progress.

There is of course the fact that if prices are rising more slowly than incomes (which may not always be the case), the purchasing power of a year's income (of 1 NAE) will be rising. I make a distinction between purchasing power, which rises or falls for everyone at the same pace, and wealth / stored / lent income which either gets repaid, repaid with interest, or repaid only in part because interest was less than AEG% p.a.

If a borrower repays less income than was lent then he/she can spend the income (NAE) that was borrowed and repay less income  / wealth, (fewer NAE), leaving him/her with a surplus of NAE AND the increased purchasing power that the income left over unpaid then has. The lender loses the same amount of NAE, AND the increased purchasing power of that lost NAE that should have been retained  Thus the wealth is shifted, not destroyed. There is a conservation of wealth here and it is defined as NAE, not purchasing power. The wealth involved in a lending contract, which is a call on the income of the borrower, is fixed at outset in NAE and the amount of wealth involved does not erode or increase, unless the true rate of interest is positive, in which case the borrower will repay more wealth than was borrowed.

The total national income (all of the NAE of a nation), comprises the NAE earned by each person added up to make the total. Each person has a share to spend, to lend, to borrow or to invest.

When it comes to mortgages, these are repaid out of incomes, so if the call on the wealth of the borrower added to the debt by a positive value of 'I%' is high, this matters. It can make a mortgage unaffordable if 'I%' is too large or if the mortgage debt (the number of NAE lent) is too large. Income (NAE) added per annum to a debt of n NAE is I% x n NAE. 

Where L = n NAE.

If either 'I%' or 'L' is too large, there is trouble ahead.

Let us say that the lender views average incomes, or NAE, as the measure of wealth that is invested. If the lender were a person and wanted to invest in something else, then among other things, there would be a choice of property and equities, both of which have a link to aggregate demand / average incomes. Most goods and services respond in price to the level of demand and so aggregate demand is, at least, loosely linked to average incomes / earnings (AEG% p.a.). Thus 'prices in general' have a link to the rate of Average Earnings Growth, AEG% p.a.

So we have now found that AEG% p.a. forms a baseline - a way to compare all investments and how fast they grow. Anything that grows as fast as AEG% p.a. preserves the NAE that was invested in it. And most investments do preserve the NAE invested in them if their price rises as fast as aggregate demand or something similar like AEG% p.a.

We can compare AEG% p.a. to the rate of capital growth of an investment and we can compare 'I%' to the dividends or the rental income obtained from such investments. The assumption is that unless some special factors are operating, (which is usually the case), rising incomes, rising AEG% p.a., will result in rising rentals and profits and rising turnover and that this will raise the value of property and Equities at the rate of AEG% p.a.

Clearly, if there was a risk free bond that rises at AEG% p.a. this would be a baseline against which all investments could be compared. I have named such bonds, 'Wealth Bonds' because they would (if they existed), preserve the income (NAE) that is saved in them, and because, if that income is not preserved, then another person will spend it - or will spend the part that was lost as I already explained and as we will see in another illustration below. 

Now think about that. If that were not true, if the total wealth (NAE) owed and repaid was not preserved, then we might not be so sure of calling this lent income a form of wealth. But if that wealth, that stored income that is lent, is going to be given in part to the borrower to spend, (if the nominal rate of interest is less than AEG% p.a.), then we must conclude that the NAE lent has been preserved, but some of it got moved from the lender to the borrower because the interest rate was less than AEG% p.a.

Repeating what I wrote earlier in new words:

When wealth is counted in NAE it is indeed something that has a character of its own. I will call that characteristic the property of conservation of wealth.

So we have a conservation law - conservation of wealth. Wealth (spendable income), as defined here, does not vanish - it gets transferred.

Looking at the total NAE that an individual gets paid or gets given one way of another in a lifetime, that can be shared with others when borrowing, and it can be increased by taking from others when lending or investing - with luck. But what does not happen - the total NAE involved in these transactions does not get destroyed. The purchasing power of all NAE can rise or it can fall but the number of NAE is what people earn in a lifetime, all added together.

Looking at an investment in goods like gold or property or equities: 

The value in NAE invested in them depends upon there being a willing (and able) buyer at the price on offer. Ability may depend upon how much people can borrow and what lending model is operating. That can change at a moment's notice. There is only a limited number of NAE out there with which to buy such investments. If someone thinks the market value is en times what the original investor (buyer) paid, that is his/her decision. The NAE paid moves to the seller. Next week that new buyer may find it can be sold for only 0.5 NAE. That unlucky or foolish person will have lost 9.5 NAE. 9 NAE went to the first buyer as a profit for that person. The other 0.5 NAE - maybe the first buyer would have made 9.5 NAE if he/she had paid only 0.5 NAE. Maybe the person he bought it from gained that 0.5 NAE. Investments are ways to capture or lose NAE from the rest of the community in accordance with whatever people think they may be work and are able and willing to pay out of their limited supply of NAE.

The wealth invested in a bond  or a loan is different. It is a call upon the income of the borrower which is clearly defined. It is safe from such things as market prices and willing buyers if it is secured by collateral or an insurance policy. But this wealth is vulnerable as it can be re-distributed if true interest rates are negative. They should not be negative, but the way things are, everything is in turmoil....

In fact, all wealth, even that invested in property  gold, equities, bonds etc is a call upon the income of a willing buyer. The money given for purchase is spendable as income and it comes either from the stored income (savings or investments) or the current income of the buyer, or some of each. So we can say that wealth in this sense, is stored spendable income.

Lenders have to compete for funds with these other investments such as property and equities. They have an advantage - their loans are secured. The risk of lending is very low and if we have a positive 'D%' p.a. built in then the level of arrears will be low too. This can be the basis for an AAA rated investment in mortgage based securities.

A ten year bond with a fixed 'I%' offered by a lender would attract a lot of money from investors wanting to get out of the uncertainties of inflation. 

A bond that has a fixed 'I%' p.a. takes the investor out of the turmoil in the economy and protects the wealth that has been invested.

Such bonds are called Wealth Bonds.

We have noted that AEG% p.a. forms a baseline for aggregate demand in an economy. GDP growth is also a form of AEG% p.a. because GDP is defined as aggregate income, like aggregate AEG when it comes to the rate of growth. This equivalence between GDP growth and AEG does not always work out exactly - the figures are not always exactly the same because of differences in definitions and in the timing of measurements and revisions to measurements, but there are strong similarities.

I did a test for South Africa's non-farm wages index and found that GDP and that wages index both rose by almost the same amount over a 25 year period, with a 5% cumulative difference. Some years the differences were greater than others. But it needs to be noted that GDP varies as the number of working people in the population changes, so immigration and emigration makes a difference, as does demographics - an aging population or a young population, can for example, affect GDP. And some economists are keenly aware that governments can manipulate GDP figures. AEG figures are not so vulnerable.


If interest equal to AEG% p.a. is added then the amount of 'average income' owed never changes. Only the addition of a marginal rate of interest called the true rate of interestI%, increases the amount of income (NAE or wealth) owed.

A ten year fixed true rate bond would attract a lot of funds from older people and their pension funds and some of their managed funds. The same applies to 15 year wealth bonds, and even 25 year wealth bonds. Some people would buy wealth bonds directly for themselves. Some wealth bonds would repay capital as well as paying interest so as to provide an annuity or to provide fund managers with a higher cash flow which they like to have. It helps them to re-balance their fund as needed. We will see how that kind of bond is the other side of a fixed true interest rate mortgage, called a Defined Cost Mortgage.


True rates are confined to a narrow band
The economics of Interest rates


Central banks have the task of managing inflation and they use interest rates to manage the level of demand for money. Because banks can leverage their lending - that is to say they can create new money by lending more than the assets that they hold, raising the rate of interest limits the amount of money that can be lent. That limits the growth of money supply.

This means that it is very important that 'I%' should be positive if there is to be a significant cost of borrowing. Otherwise the demand for loans will exceed the supply, or the demand for loans will be so great as to force the Central Bank to increase interest rates to prevent an inflationary spiral caused by run-away borrowing and credit creation.

Imagine being able to borrow at AEG% interest and then invest in property at AEG% p.a. growth and get rental on top - for example. Everyone would start doing that eventually. The whole economy would run wild and everything would collapse in hyper-inflation.

At the same time, 'I%' must not be too high for too long because it will make lending produce a secure return that is greater than that which can be obtained by investing in property or equities and without the risk. The result will be a flood of money that is too expensive to lend. Borrowers will be unable to afford such an expense.

So one way or another market forces keep true interest rates in a very narrow band, except when they wander off course for a while in both directions. The reasons for that wandering is the subject of conjecture at this time, but instinctively one knows that it is caused by the instabilities in economies, instabilities largely caused by the debt structures themselves as currently operating. And then there is the over-response and unbalanced responses made by Central Banks and their unbalanced instruments for intervention.

There can also be long term imbalances caused by an imbalance of trade and hoarding of a foreign currency.

My ambition is to show how all of these imbalances and overshoots can be minimised - much minimised. I think it can be done. Even the currency issue and balance of trade issues.

These are things I write about and discuss online and elsewhere when discussing monetary policy in the new and the old environment, before and after these new debt structures and other measures relating to trade and money creation, (another topic for another time and place), are introduced.

Studies done on the UK, and other places (but mainly the UK) suggest that over a business cycle, on average,  'I%' has to lie somewhere between 2% and 4% unless the Central Bank has distorted interest rates or other factors have done so. For example Quantitative Easing or an excess of foreign money coming in can lower interest rates for several years. Such distortions can greatly extend the business cycle which is why I chose to say the average rate of I% will tend to lie in this narrow range 'over a business cycle'. 

Another distorting factor when raising the interest rate would precipitate a housing crisis or an economic slowdown. That is what happens with the current mortgage models. The first  scenario (the crisis) is what the Fed discovered by trying to raise interest rates enough to curb inflation after a period of super-low interest rates; and the second scenario (a failure to raise the interest rate enough to slow inflation) is what the Bank of England discovered, making them reduce interest rates and miss the inflation target in circa 2004 - I have the charts. I may add them later.

The problem arises because if the Fed had had its way the monthly payments cost 'P p.a.' of USA mortgages would have risen by 54% on 25 year mortgages and by 58% on 30 year mortgages and by an enormous percentage on interest only mortgages! Not much like a rental or any other pricing response to an increased rate of aggregate demand, is it?

Mortgage costs and sizes are over-sensitive to interest rate changes as things are currently done. The AEG% component is not separated out. That is one reason. Another reason is that all of the interest is demanded at all times and none can be rolled over to smooth things out.

Such cost changes that exceed the AEG baseline of other price changes such as rentals and goods and services change spending patterns as well as wealth patterns (as property values bubble and crash) also change the whole spending pattern of the nation. When spending shifts from one sector to another like that it destroys jobs. 

When the wealth invested in property and other investments are highly sensitive to interest rates as interest rates rise and fall, and with all of that alternating in direction, that enlarges the business cycle significantly. This is a significant barrier to the achievement of sustainable economic growth.

As we will see later, with the ILS Model and risk management methods, these problems of rapidly changing mortgage costs and mortgage sizes (loan/income ratios) can be removed. Goodbye to at least some of the interest rate based distortions in our economies.

The estimated upper true interest rate bound of 4% is for the UK and the USA where banking has been efficient. It may be more costly in future so watch out - BASEL III etc to blame here. This is because the long term rate of return on Equities has been around 7% p.a. in real terms. See the study on Siegel's Constant and other studies that came up with similar figures over 200 years.

This true rate of return on equities may be less than 5% p.a. - that is less than 5% p.a. above AEG% p.a. The difference between real and true rates of return is because incomes rise faster than prices most of the time. I will come back to that a bit later to make sure this difference is clear and well understood. I also go into that in detail on the Siegel's Constant page just mentioned.

It means that if an investment keeps pace with AEG it also keeps pace with real economic growth and it needs to do that to preserve the wealth that has been invested - to stop some of it escaping to be spent by others. 

Distorting interest rates leads to unsustainable imbalances in the economy. It redistributes wealth and does harm. Readers should bear this in mind as they read further because my new debt structures should remove the need for distorted interest rates, which are usually distorted (held down as now with QE or by Central Bank policy),  to reduce the rate of default on debts (as just mentioned) which mainly occur in the housing sector insofar as this sector is the most disturbed by interest rate increases and decreases.

Currently there is also the fear of undermining property values. Both of these issues (of excessive interest rate sensitivity) form a barrier to the Central Bank's interest rate policy.

If we can manage to find a way of keeping D% positive even as interest rates are rising, then interest rates can be free to rise without causing arrears. The exciting part is that tests show that we can do that.

The secret is that for every 1% increase in the true interest rate D% has to fall by 1% if P% and C% are not to be disturbed: 

P% = C% + D% + I%

So if the true rate can only rise by 2% at most from it long term average rate, then D% can always be at least 2% positive if it starts out as 4% for a new mortgage. Short term extremes in I% can be smoothed out by the averaging process as we will see in the illustrations to come.

In short, the other exciting thing that I discovered after much analysis and careful thinking was that, with these new debt structures in place, major economic interventions and such distortions need not happen and should not be tolerated. When we take out the distortions that are caused both directly and indirectly in this way by the debt structures which are currently in use, the economy should behave well and such interventions (for example also a Keynesian stimulus) should not need to be large, even if one is actually needed.

An economy that has secure borrowing and secure wealth with stability in its finances, should be an economy in which almost everyone wants to spend more to get more and so everyone starts to earn more as they provide more for one another. That is how it works eventually, but with financial stability, that is how it should work all the time - other than after currency crises and natural disasters.

I say that a stimulus may not be needed in this case after careful consideration of how business cycles get amplified by a combination of the instability of mortgage finance and bond values as interest rates change. This is then further amplified by the 'wealth effect' and a general lack of confidence which results. Everything reverses including the wealth (greed and panic) effect when property prices and bond values shoot upwards again. This sets the scene for the next crisis as interest rates start rising once more.

If we can get rid of these amplifications and distortions, then I% should also be stable. But in the meantime we have to deal with unstable I% and we have to be sure that the new mortgage model can cope with that. The outcome of my detailed studies shows that this is all quite possible.

We saw that if the true rate of interest is very low it will cause over-lending and inflation. And if the true interest rate is kept above around 4% for very long, it will result in such a high demand for deposits and bonds by investors that lenders will be unable to lend the money on deposit and in bonds. Free markets do not permit this kind of thing.

We noted that if the true interest rate is higher than the long term rate of return on equities then a 10 year true rate bond issue would be over-subscribed many times. Lenders could not lend that much at that cost to borrowers.

Thus true interest rates are largely confined to a narrow range over the medium to long term, which my historic data seems to confirm. I WILL PROVIDE THE CHARTS FOR SOME OF THESE STATISTICS ON ANOTHER PAGE SHORTLY. And historic data for a number of nations can be found on my interactive spreadsheets that are for sale. 

Spreadsheet owners can run any number of tests on the data and many fascinating tests are suggested. One test shows that if interest rates and AEG rates rise every year by 1% p.a. starting at 3% interest and 0% AEG, the ILS Mortgage model does not really notice that happening. It is not at all disturbed! 

Whatever the rate of AEG% p.a. is doing, if I% p.a. is fixed then D% p.a. is fixed and you get what I call a Defined Cost Mortgage (fixed I% and fixed D%) like this:

The shapes and slopes do not alter with varying AEG% p.a. not with external rates of interest. Users can put any figures into my interactive spreadsheets and see what happens to both the ILS and the traditional mortgage models. Results are displayed on ten different Bar Charts as well as full tabulations.

Interest rate distortions were mentioned, which relate to the instability of mortgage payments as currently constructed. For example the current practice of linking the amount lent to the nominal rate of interest results in a great deal of instability in  the loan/income ratio (mortgage sizes), and so affecting property values and interest rate changes hugely increases the risk of borrowing arrears when variable rates are used.  These distortions to interest rates and property values feed into the economy and get amplified by the wealth effect making everything unstable.

Let's get back to the maths now:

SORRY - I am still editing this web page and what follows has already been covered. I will am now removing the relevant parts. This may take time and it may leave a temporary mess. Please come back later...

Delete this red font - message to me
Maybe keep the blue font or use it in the final text

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